Mathematics Extension 2

Transcription

1 S Y D N E Y B O Y S H I G H S C H O O L M O O R E P A R K, S U R R Y H I L L S 005 HIGHER SCHOOL CERTIFICATE TRIAL PAPER Mthemtics Extension Generl Instructions Totl Mrks 0 Reding Time 5 Minutes Attempt questions 8 Working time Hours Write using blck or blue pen. Pencil my be used for digrms. Bord pproved clcultors mybe used. Ech Section is to be returned in seprte bundle. All necessry working should be shown in every question. Exminer: C.Kourtesis NOTE: This is tril pper only nd does not necessrily reflect the content or formt of the finl Higher School Certificte exmintion pper for this subject. SHS 005 Extension Tril HSC Pge

2 Section A (Strt new nswer sheet.) Question. (5 mrks) Evlute 4 + x dx. 0 Mrks (b) Find cos xsin 4 x dx. (c) Use integrtion by prts to find te t dt. (d) Find rel numbers nd b such tht Hence find x( π x) = x + b π x. dx. x( π x) (e) Evlute ( x )dx. (f) Use the substitution x = t to prove tht Hence evlute f (x)dx = f ( x)dx. 0 π 0 0 log e ( tn x)dx SHS 005 Extension Tril HSC Pge

3 Question. (5 mrks) (b) (c) If z = + i nd w = + i find Im( z w). On n Argnd digrm shde the region tht is stisfied by both the conditions Re(z) nd z. If z = nd rg z = θ determine Mrks i i rg z z (d) If for complex number z it is given tht z = z where z 0, determine the locus of z. (e) A complex number z is such tht rg( z + )= π 6 nd rg ( z )= π. Find z, expressing your nswer in the form + ib where nd b re rel. (f) The complex numbers z, z nd z re represented in the complex plne by the points P, Q nd R respectively. If the line segments PQ nd PR hve the sme length nd re perpendiculr to one nother, prove tht: z + z + z = z ( z + z ) SHS 005 Extension Tril HSC Pge

4 Section B (Strt new nswer sheet.) Question. (5 mrks) If i is zero of the polynomil z + pz + q where p nd q re rel, find the vlues of p nd q. Mrks (b) If α, β nd γ re roots of the eqution x + 6x + = 0 find the polynomil eqution whose roots re αβ, βγ nd αγ. (c) Consider the function f (x) = x + 4 x. Show tht the curve y = f (x) hs minimum turning point t x = 4 nd point of inflexion t x = 6. Sketch the grph of y = f (x) showing clerly the equtions of ny symptotes. 5 (d) Use mthemticl induction to prove tht n! > n for n > where n is n integer. SHS 005 Extension Tril HSC Pge 4

5 Question 4 (5 mrks) If f (x) = sin x for π x π drw net sketches, on seprte digrms, of: y = f ( x) y = f x + π (iii) y = f (x) (iv) y = f( x ) (b) Show tht the eqution of the tngent to the curve x + y = t the point P( x 0, y 0 ) on the curve is xx + yy =. o 0 (c) Consider the polynomil P(x) = x 5 x +. By considering turning points on the curve y = P(x), prove tht P(x) = 0 hs three distinct roots if > SHS 005 Extension Tril HSC Pge 5

6 Section C (Strt new nswer booklet) Question 5 (5 mrks) A prticle of mss m is thrown verticlly upwrd from the origin with initil speed V 0. The prticle is subject to resistnce equl to mkv, where v is its speed nd k is positive constnt. Mrks (iv) Show tht until the prticle reches its highest point the eqution of motion is &&y = ( kv + g) where y is its height nd g is the ccelertion due to grvity. Prove tht the prticle reches its gretest height in time T given by kt = log e + kv 0 g. If the highest point reched is t height H bove the ground prove tht V 0 = Hk + gt. 4 4 (b) If α nd β re roots of the eqution z z + = 0 find α nd β in mod-rg form. show tht α n + β n = n+. cos nπ 4. SHS 005 Extension Tril HSC Pge 6

7 Question 6 (5 mrks) A group of 0 people is to be seted t long rectngulr tble, 0 on ech side. There re 7 people who wish to sit on one side of the tble nd 6 people who wish to sit on the other side. How mny seting rrngements re possible? (b) The re enclosed by the curves y = x nd y = x is rotted bout the y xis through one complete revolution. Use the cylindricl shell method to find the volume of the solid tht is generted. (c) The digrm shows hemi-sphericl bowl of rdius r. The bowl hs been tilted so tht its xis is no longer verticl, but t n ngle θ to the verticl. At this ngle it cn hold volume V of wter. The verticl line from the centre O meets the surfce of the wter t W nd meets the bottom of the bowl t B. Let P between W nd B, nd let h be the distnce OP. r Explin why V = π( r h )dh. r sinθ Hence show V = r π ( sinθ + sin θ). (d) Show tht x 4 + y 4 x y. If P(x, y) is ny point on the curve x 4 + y 4 = prove tht OP 4, where O is the origin. SHS 005 Extension Tril HSC Pge 7

8 Section D (Strt new nswer booklet) Question 7 (5 mrks) How mny sets of 5 qurtets (groups of four musicins) cn be formed from 5 violinists, 5 viol plyers, 5 cellists, nd 5 pinists if ech qurtet is to consist of one plyer of ech instrument? (b) If t = tnθ, prove tht tn 4θ = 4t ( t ) 6t + t. 4 If tnθ tn 4θ = deduce tht 5t 4 0t + = 0. (iii) Given tht θ = π 0 π nd θ = re roots of the eqution 0 tnθ tn 4θ =, find the exct vlue of tn π 0. 4 (c) C M A N B Two circles intersect t A nd B. A line through A cuts the circles t M nd N. The tngents t M nd N intersect t C. 5 Prove tht CMA + CNA = MBN. Prove M, C, N, B re concyclic. SHS 005 Extension Tril HSC Pge 8

9 Question 8 (5 mrks) 6 The digrm bove shows the grph of y = log e x for x n +. By considering the sum of the res of inner nd outer rectngles show tht Find ln x dx. n+ n+ ( n ) < x dx < ( n + ) ln! ln ln! (iii) Hence prove tht e n > ( n + )n n! (b) If root of the cubic eqution x + bx + cx + d = 0 is equl to the reciprocl of nother root, prove tht + bd = c + d. This question continues on the next pge. SHS 005 Extension Tril HSC Pge 9

10 (c) A stone is projected from point O on horizontl plne t n ngle of elevtion α nd with initil velocity U metres per second. The stone reches point A in its trjectory, nd t tht instnt it is moving in direction perpendiculr to the ngle of projection with speed V metres per second. Air resistnce is neglected throughout the motion nd g is the ccelertion due to grvity. If t is the time in seconds t ny instnt, show tht when the stone is t A: 6 V = U cotα U t = gsinα. This is the end of the pper. SHS 005 Extension Tril HSC Pge 0

11 STANDARD INTEGRALS x n dx = x n+, n ; x 0,if n < 0 n + dx = ln x, x > 0 x e x dx = ex, 0 cos xdx = sin x, 0 sin xdx = cos x, 0 sec xdx = tn x, sec x tn x dx = sec x, 0 + x dx x dx x dx = tn x, 0 = sin x, > 0, < x < ( ), x > > 0 ( ) = ln x + x dx = ln x + x + x + NOTE: ln x = log e x, x > 0 SHS 005 Extension Tril HSC Pge